Remark 3.2 (Static condensation for \(k\ge 2\) ) 

When polynomial degree \(k\ge 2\) , we shall
solve the linear system problem
(REF ) using static condensation to locally eliminate
interior velocity DOFs and high-order pressure DOFs
[1]}.
The resulting global linear system after static condensation
consists of DOFs for the normal component of velocity
approximation (of degree \(k\) )
in \(\hbox{$V$}_{h,0}^k\) 
and the tangential (hybrid) velocity approximation (of degree \(k-1\) )
in \(\widehat{\hbox{$V$}}_{h,0}^{k-1}\) 
on the mesh skeleton (facets), and
cell-average of pressure approximations (of degree 0) on the mesh.
We denote the compound velocity space corresponding to
the DOFs on mesh skeleton by \(\underline{\hbox{$V$}}_{h,0}^{k,\mathsf {gl}}\)  which is a subset of the compound space
\(\underline{\hbox{$V$}}_{h,0}^k\) . The global pressure space is
simply the space of piecewise constants \(Q_h^0\) .
The linear system after condensation has a similar structure as that in
(REF ) with a reduced size. We shall apply the precondioned
MinRes solver for the condensed system in this case.
For this case, the matrices \(\mathsf {A}\) 
and \(\mathsf {S}\)  shall be understood to be defined on the
reduced spaces \(\underline{\hbox{$V$}}_{h,0}^{k,\mathsf {gl}}\)  and \(Q_h^{0}\) , respectively.

